Randomness in the evacuation route selection of large-scale crowds under emergencies

Applied Mathematical Modelling xxx (2015) xxx–xxx

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Randomness in the evacuation route selection of large-scale crowds under emergencies Jinghong Wang a,b, Jinhua Sun b,⇑, Siuming Lo c a

Jiangsu Key Laboratory of Urban and Industrial Safety, College of Urban Construction and Safety Engineering, Nanjing Tech University, Nanjing 210009, China State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei 230026, China c Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong b

a r t i c l e

i n f o

Article history: Received 5 June 2013 Received in revised form 3 December 2014 Accepted 12 January 2015 Available online xxxx Keywords: Large-scale evacuation Stochastic Markov model Route selection Psychological risks Physiological risks

a b s t r a c t To date, considerable research has been conducted into micro and macro evacuation models, fuelling the development of the field of emergency evacuation. However, the uncertainties of large-scale evacuation under emergencies have not yet been fully elucidated. In this paper, the impact of the crowd physiological and psychological factors on large-scale evacuation is presented and quantified, then incorporated into a random Markov route selection model of evacuees, to obtain a probabilistic description of crowd route selection. Against a background of instantaneous leakage and diffusion of CO poison gas, a detailed analysis of the uncertainties inherent in the evacuation process and results (specifically, the sensitivity of clearance time to a variety of factors) is conducted using the Markov process, integrating the initial distribution of crowds, route travelling capability, and so on. The variation rule of clearance time under the influence of relevant parameters is revealed and also the significance of considering the psychological and physiological risks in large-scale emergency evacuation is proved.  2015 Published by Elsevier Inc.

1. Introduction When natural disasters such as earthquakes, tsunamis and hurricanes are imminent or have recently occurred, often a large number of people in the affected regions require evacuation. When man-made disasters occur, such as terrorist attacks in a metro station or riots in a major demonstration, emergency evacuation of the large crowd present is usually also required. Industrial accidents, such as hazardous material leakage and explosions, also tend to cause a large number of surrounding residents to be evacuated. Such emergency evacuations, which involve a large number of people and differ from the escape actions of small or medium-sized crowds in ordinary building evacuations, are described as large-scale evacuations. For such an evacuation, although existing micro and macro evacuation models have achieved effective analysis of personnel movement or the overall traffic situation during evacuation [1–4], and thus played an important role in predicting evacuation time and evaluating evacuation plans, in emergencies, people are affected by abnormal pressure, thus individual and group psychological and behavioral reactions are quite different from those in normal situations [5] and can cause many uncertainties.

⇑ Corresponding author at: 96 Jinzhai Road, Hefei, Anhui, Postal code: 230026, China. E-mail address: [email protected] (J. Sun). http://dx.doi.org/10.1016/j.apm.2015.01.033 0307-904X/ 2015 Published by Elsevier Inc.

Please cite this article in press as: J. Wang et al., Randomness in the evacuation route selection of large-scale crowds under emergencies, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.033

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A great many factors, such as the disaster environment, guidance information, risk perception of the individual and individual psychology, can influence individual behavioral decisions in the evacuation process, and the impact of these factors changes with time. Hoogendoorn and Bovy [6,7] stated that individual behavior is affected by external factors, such as environmental stimuli and obstructions, internal factors, such as pedestrian intention and time constraints, and traffic conditions, such as average speed and congestion. Therefore, pedestrian behavior is uncertain. Given this uncertainty, they proposed a route selection model able to determine the continuous movement path of pedestrians and make optimal dynamic decisions as to resource allocation in continuous space and time. In this model they considered the behavioral cost of physical space restrictions and pedestrian dynamics. Sime [8] presented a pedestrian evacuation time analytical model, in which the impact of multiple factors, such as pedestrian psychology, building structures and evacuation facilities, on pedestrian route choice behavior was considered. In large-scale evacuation under emergencies, the individual is more significantly affected by the disaster environment, population density and emotional state than in conventional cases. Thus, the behavioral uncertainty is greater. Escape speed and path selection are real-time varying, and depend on the perception of the surrounding environment, and physiological and psychological state of the individual [9,10]. Therefore, in addition to objective factors, such as disaster environment, time cost and congestion, we should also consider subjective factors relating to the crowd’s physiology and psychology while studying the uncertainties in large-scale evacuation, so as to reflect more realistically the stochastic characteristics of crowd route selection behavior under emergencies. According to their different principles of route selection, regional evacuation models may be divided into three categories. (1) Evacuation models which employ pre-established evacuation routes. These models usually determine routes before simulation according to an existing evacuation plan or principles determined by the model users, such as the nearest destination [11]. (2) Evacuation models which employ route optimization. At the core of these models is the optimization objective of the evacuation route. Different optimization objectives will result in selection of different routes [12]. Most models design and simulate the evacuation process according to the principles of least evacuation time or shortest distance. (3) Evacuation models which dynamically select evacuation routes according to certain principles, which usually include:  Evacuation in the direction with lowest disaster risk, in which disaster spread direction, spread speed, etc., should be considered.  Evacuation towards the destination with least transit cost (time or distance).  Evacuation along paths where congestion is less, in which real-time traffic information should be integrated.  Evacuation according to final destination or vehicle type. Depending on the guiding principle(s), the evacuation route selection may be either deterministic [13,14] or probabilistic [15]. To describe uncertainties, stochastic models based on the Markov approach have been widely applied in the field of traffic flow [16,17], but these models ignore subjective characteristics and only focus on objective information related to traffic network, such as distance and the congestion. In this work, the primary concern is large-scale crowd evacuation in which the physiological and psychological factors of the crowd itself are significant, affecting not only the objective parameters, e.g. moving speed or crowd density, but also other subjective aspects, such as whether people can withstand movement through a dangerous environment (such as a toxic gas cloud), whether panic will arise and influence movement, etc. In consideration of these problems, the stochastic Markov model of large-scale crowd evacuation presented in this work will not only contain the key features of prior work, but also some original macro elements related to crowd physiology and psychology. 2. Probabilistic description of route selection 2.1. The basic stochastic Markov model Markov process-based random evacuation is essentially a process of crowd density changing with time. In order to calculate the changes in population distribution in the evacuation space, a dynamic model based on the Markov random process is used to simulate the movement of the population. First a schematic diagram of the evacuation area is established, following the approach used in prior literature [17]. The key elements are as follows: (1) The area to be evacuated is divided into N sub-areas. Each sub-area is known as a grid node. (2) Each node represents an artificially delineated space region, whose size and shape may be changed as required. In other words, a node does not necessarily need to correspond to a single building or functional area in the real world. It may represent one building or a region consisting of multiple buildings. (3) The nodes are connected by line segments. If there are two nodes representing two buildings, the connection between them can represent a road between the buildings. If the two nodes represent multiple buildings, the connection can represent the link capability between them. By assigning the parameters of travelling capacity and length to links, the link capability of the connections may be characterized. For a n  n grid network, the corresponding number of links is (1 + 2 +    + n)  4 = 2n  (n + 1). Please cite this article in press as: J. Wang et al., Randomness in the evacuation route selection of large-scale crowds under emergencies, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.033

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The key stage in constructing a Markov chain is the definition of the state set. Therefore, we define that the stochastic process of the Markov chain for mass evacuation is that the crowd move from one node to the next via random selection of the next node, in which the probability of choosing the next node depends on a number of factors. In the basic model, the first consideration is the cost (distance or time cost) of migration to each target node. We define aij(t) as the probability of the crowd from node i at time t moving to node j.

aij ðt Þ ¼

U ; dij

ð1Þ

where U is the average speed of the crowd, and dij is the distance between node i and node j. When there is no connection or the connection between node i and node j is constrained, then aij(t) = 0. It may be observed that when the crowd chooses between different destinations, a node with a shorter transition time has a higher probability of being selected, but we assume that the road network is safe and the crowd is familiar with the road network, or there is appropriate evacuation guidance which can inform the crowd of the nearest destination. 2.2. Amendments to the basic model Previous researchers have considered a range of factors when amending the basic stochastic Markov model, for example, whether there is a road network, whether vehicles are used and the vehicle type, and the real-time traffic information in specific areas, i.e. concerning the congestion level and the weather [18]. Here we only consider pedestrian evacuation. Therefore, those factors related to vehicle traffic are not considered in our model. Taking the outcomes of prior research into consideration, we further amend the basic stochastic Markov model by incorporating some emphasis on crowd physiological and psychological factors. 2.2.1. Consideration of congestion In the basic model, the calculation of aij(t) depends upon the condition that the crowd can move smoothly over the road network. Thus, when the average speed is a constant, aij(t) will only depend on the distance between nodes. However, if the crowd density is large and there is congestion along the links, aij(t) will not only depend on distance, but also on the congestion level, which reflects the accessibility of alternative target nodes. When the congestion level of the link between node i and node j increases, then aij(t) will decrease, and vice versa. aij(t) is also affected by the congestion level of the source nodes. The higher the congestion level of a source node, the greater the difficulty for a crowd seeking to pass from the source node to the target node. Therefore, we use the following formula to reflect the influence of congestion level on aij(t):

  8U ½1  r i ðt Þ 1  r j ðt Þ ; i–j > > < dij N X aij ðtÞ ¼ ; > aij ðt Þ; i¼j > :

ð2Þ

i–j;j¼1

where ri(t) represents the congestion level at node i. Numerous traffic simulation researchers have described vehicle congestion using an exponential function [19]. Adapting the methods of such research, and given the similarity between vehicle and pedestrian flow, we define the crowd congestion level as follows:

( r i ðt Þ ¼

r max ; Pi ðt Þ P cM i h  i ; Pi ðt Þ r max 1  exp M P ðtÞ ; Pi ðt Þ < cMi i

ð3Þ

i

where Pi(t) represents the number of evacuees at node i at time t, and the parameter Pi(t) may be calculated by Pi(t) = xi(t)PT, in which xi(t) is the probability that an evacuee participating in the evacuation is at node i at time t; rmax and c are two parameters related to the congestion level, which are both less than 1; and Mi denotes the maximum capacity of node i. It may be observed that the congestion level of node i will increase exponentially with an increase of population at this node, before reaching a state of overcrowding, when the congestion level will peak at rmax. Fig. 1 illustrates the change in congestion level as population at the node increases, when Mi equals 30, 40 or 50, rmax = 0.9 and c = 0.8. Our results suggest that, given equal crowd populations, nodes with larger maximum capacity will experience lower congestion levels than those with smaller maximum capacity, before reaching their maximum congestion levels. 2.2.2. Consideration of crowd physiological factors When facing hazardous conditions, such as toxic gas diffusion or regional fire spread, the ability of the evacuating crowd to take actions will be affected by harm from the toxic gas, heat or smoke, which may even result in death. To facilitate analysis, an accidental toxic gas leak will be used as the disaster background and the basic model will be amended as follows, in consideration of crowd physiological factors. The damage range of toxic gas may be divided into four broad zones according to [20], namely:  The death zone, in which people must be safeguarded and promptly evacuated, otherwise approximately 50% may die from poisoning.

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1.0 0.9

Mi=30

0.8

Mi=40

0.7

Mi=50

ri(t)

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

5

10

15

20

25

30

35

40

45

Pi(t) Fig. 1. The change in congestion level with the number of population at a given node.

 The serious injury zone, in which severe or moderate poisoning will occur, and some individuals may die from poisoning if not hospitalized.  The injury zone, in which most people will suffer mild poisoning or exhibit inhalation reaction symptoms, but may be rehabilitated through outpatient treatment.  The inhalation reaction zone, in which some individuals will exhibit inhalation reaction symptoms, but can usually be rehabilitated within 24 h of evading the toxic gas. Once the toxic gas type, gas concentration and contact time have been confirmed, the boundaries of the damage zones may be calculated according to the poison dosage. The boundary of the death zone may be determined from the median lethal dose of the toxic gas to humans, while the boundaries of the serious injury and injury zones are determined according to the median injury dose and the median toxic dose respectively. The boundary of the inhalation reaction zone is determined according to the maximum enduring concentration of a given toxic gas in the human body. Haber [21] et al. stated that the product of toxic gas concentration C and the maximum time t which humans can endure at this concentration is a constant, i.e.

W ¼ C  t:

ð4Þ

It should be noted that when the toxic gas concentration is low, because inhalation and exhalation of toxic gas are conducted simultaneously, the enduring time t is in fact larger than the result calculated by the above formula. When the toxic gas concentration is higher, Purser [22] confirmed through biological experiment that the concentration of CO gas and the enduring time of monkeys approximately satisfy this formula, and the constant W is equal to 27,000 ppm  min. To facilitate analysis, we take CO gas as an example and adopt the value of W established by Purser. If we know the toxic gas concentration at the nth time step is Cn, then the maximum time for which people can endure such a concentration is

t ¼ W=C n :

ð5Þ

Thus, we define a parameter a to represent the damage extent of toxic gas to the human body within Dt:

a ¼ Dt=ðW=C n Þ:

ð6Þ

From this definition, if a P 1, then humans cannot endure the harmful effects of the toxic gas and cannot maintain normal movement, thus aij(t) = 0, or close to 0. If a < 1, then humans can endure the harmful effects of the toxic gas for Dt. We further assume that when a approaches 1, i.e. the human body approaches its tolerance limits, aij(t) approaches 0; when a equals 0, i.e. the toxic gas has no harmful effect on the human body, then aij(t) is equal to its normal value when no toxic gas is present. From these hypotheses, we obtain

aij ðtÞ ¼

U 10a U 10DtCn e ¼ e W ; dij dij

ð7Þ

which reflects the amendment of the basic stochastic model in consideration of the physiological risk caused by toxic gas effects. Thus, by integrating congestion level and crowd physiological risk, we have

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8 U 10DtCn    e W ½1  r i ðtÞ 1  r j ðt Þ ; i–j > > < dij N aij ðtÞ ¼ : X > > aij ðt Þ; i¼j :

5

ð8Þ

i–j;j¼1

2.2.3. Consideration of crowd psychological factors More recently, Cooper et al. [23] suggest that there are three governing psychological factors which may affect crowd movement, one of which is that people will walk at a maximum speed dependent on certain environmental conditions. Thus, in investigating crowd psychological factors in this work, our primary focus is panic in mass evacuations. Generally, the panic of a crowd is closely related to the disaster environment. People in the most severely affected area are thought to be more vulnerable to panic. In addition, some studies have demonstrated that the more a crowd panics, the higher the expected evacuation speed will be when all other parameters remain constant. In other words, the crowd will be more likely to rush to escape [24]. According to the development of a disaster and its sphere of influence on the evacuation routes, the evacuation routes may be divided into three types, namely ideal evacuation routes, feasible evacuation routes and escape evacuation routes [25,26]. Maintaining these divisions and considering crowd panic, in this work we divide and describe the evacuation routes under the influence of toxic gas as follows:  Ideal evacuation routes, which are generally located in or outside the inhalation reaction zone and without evident effects of toxic gas diffusion. People along these evacuation routes are generally in an emotionally stable state and capable of rational evacuation without panic.  Feasible evacuation routes, which are generally located in the serious injury or injury zones, and are affected by toxic gas diffusion, but without serious or life-threatening harm. People in these evacuation routes can generally maintain rational evacuation.  Escape evacuation routes, which are located in the death zone: toxic gas concentration along these routes has reached the maximum endurance level of the human body. Those who fail to escape from the death zone will suffer life-threatening damage. Therefore, people along these routes will be prone to obvious panic due to the high threat level and urgency of evacuation. Accordingly, we propose a correction coefficient c for the evacuation speed caused by different degrees of panic, namely:

c = 1, for ideal evacuation routes. c = 1.3, for feasible evacuation routes. c = 1.8, for escape evacuation routes. Let us assume that the mass evacuation speed is 90 m/min (1.5 m/s) under normal circumstances. Then based on the above correction, the evacuation speed along feasible evacuation routes and escape evacuation routes will be 117 m/min (1.95 m/s) and 162 m/min (2.7 m/s) respectively. It should be noted that the correction of evacuation speed along different evacuation routes does not consider the ‘‘faster is slower’’ phenomenon, because the correction due to congestion level has already been considered. In addition, evacuation speed here refers to the average speed of the crowd as a whole, without considering the influence of individual differences on movement speed. When the amended transition probability rate aij(t) is integrated with consideration of the congestion level, the crowd physiological and psychological factors will be:

8 cU 10DtCn    e W ½1  r i ðt Þ 1  r j ðt Þ ; i–j > > < dij N aij ðtÞ ¼ ; X > > aij ðt Þ; i¼j :

ð9Þ

i–j;j¼1

where ri(t) is calculated by Formula (3). 2.3. The Markov process of crowd randomness evacuation As was stated above, Markov process-based random evacuation is essentially a process of crowd density (the distribution of the crowd in space) changing with time. To facilitate subsequent analysis, some basic definitions and related theorems of Markov chains are first introduced. Definition 1. Suppose the state space of a stochastic sequence {Xn; n P 0} is I. If for "n 2 N0 and i0, i1, . . . , in, in+1 2 I, P{X0 = i0, X1 = i1, . . . , Xn = in} > 0, there is:

PfX nþ1 ¼ inþ1 jX 0 ¼ i0 ; X 1 ¼ i1 ; . . . ; X n ¼ in g ¼ PfX nþ1 ¼ inþ1 jX n ¼ in g;

ð10Þ

then this stochastic sequence{Xn; n P 0} is a Markov chain. Please cite this article in press as: J. Wang et al., Randomness in the evacuation route selection of large-scale crowds under emergencies, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.033

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This definition describes the characteristics of the Markov chain, i.e. when current knowledge or information is given, the past (namely the previous history state before the current state) is irrelevant for predicting the future (namely the future state after the current state). This feature is also known as Markov property or so-called ‘‘no aftereffect’’. Definition 2. Suppose {Xn; n P 0} is a Markov chain, and the state space is I. For "i, j 2 I, we describe

PfX nþ1 ¼ jjX n ¼ ig¼p ^ ij ðnÞ;

ð11Þ

as the one-step transition probability of the Markov chain {Xn; n P 0} at time n, and it satisfies

8 < 0 6 pij ðnÞ 6 1; i; j 2 I X pij ðnÞ ¼ 1; i 2 I : :

ð12Þ

j2I

If for "i, j 2 I, there is

PfX nþ1 ¼ jjX n ¼ ig¼p ^ ij ðnÞ  pij ;

ð13Þ

i.e. the right section of the above formula is independent of time n, then this Markov chain is considered a homogeneous Markov chain. Theorem 1. If {Xn; n P 0} is a homogeneous Markov chain and its state space is I, then:

PfX 0 ¼ i0 ; X 1 ¼ i1 ; . . . ; X n ¼ in g ¼ PfX 0 ¼ i0 gPfX 1 ¼ i1 jX 0 ¼ i0 g  P fX 2 ¼ i2 jX 1 ¼ i1 g . . . P fX n ¼ in jX n1 ¼ in1 g:

ð14Þ

This implies that the joint distribution of X0, X1, . . . , Xn may be determined from the initial distribution and the one-step transition probability, i.e. n Y PfX 0 ¼ i0 ; X 1 ¼ i1 ; . . . ; X n ¼ in g ¼ p0 ðiÞ pik1 ik :

ð15Þ

k¼1

Theorem 2. If {Xn; n P 0} is a Markov chain and its state space is I, then for any given n integers 0 6 k1 < k2 <    < kn, there is:

    P X kn ¼ ikn ; X kn1 ¼ ikn1 ; . . . ; X k1 ¼ ik1 ¼ P X kn ¼ ikn jX kn1 ¼ ikn1 ;

ð16Þ

i.e. the sub-chain of a Markov chain is also a Markov chain. Using Markov process theory, we can have

x_ ðt Þ ¼ xðt Þ  Aðt Þ;

ð17Þ

_ where x(t) is a 1  N vector with elements [xi(t)]; 0 6 xi(t) < 1; xðtÞ is a 1  N vector with elements ½x_ i ðtÞ; A(t) is a N  N vector with elements [aij(t)] (can be regarded as the state-transition probability rate); [xi(t)] is the probability that an evacuee is at node i at time t; and ½x_ i ðtÞ is the derivative of [xi(t)] with time. Since aij(t) is a function of xi(t), the Eq. (17) can also be written as

x_ ðt Þ ¼ xðt Þ  A½xðtÞ:

ð18Þ

To solve the Eq. (18) numerically, the time of transitions is discretized as follows:

x½ðn þ 1ÞDt  ¼ xðnDt Þ  GðnDt Þ;

ð19Þ

where G(nDt) = /[(n + 1)Dt, nDt] represents the state-transition matrix of /ðt; t 0 Þ between time (n + 1)Dt and nDt. Here, / (t, t0) is the state-transition matrix of the original continuous system given by Eq. (17). Using simplified symbols x(n) to represent x(nDt), then Eq. (19) can be described as:

xðn þ 1Þ ¼ xðnÞ  probðnÞ;

ð20Þ

where x(n) is a N  N vector with elements xi(n); xi(n) is the probability that an evacuee participating evacuation is at node i during time interval Dtn (between time tn and tn + Dtn, n = 1, 2, . . . , T); prob(n) is a N  N vector with elements pij(n). pij(n) is the state-transition probability of a person from node i at time tn = n  Dt to node j at time tn+1 = (n + 1)  Dt, which satisfies

8 0 6 pij ðnÞ 6 1 > > < N X ; > pij ðnÞ ¼ 1 > :

ð21Þ

i;j¼1

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in which n is the current time step (n = 1, 2, . . . , T); T is the total number of time steps, T = Te/Dt; Te is the total evacuation time; and Dt is the time interval (1 min in this work). Therefore, the state-transition probability pij(n) will be:

" !# 8 N X > aij ðt Þ > > P i–j 1  exp  aij ðt Þ  Dt > > < Ni–j;j¼1 aij ðtÞ i–j;j¼1 ! pij ðnÞ ¼ ; N > X > > > exp  i ¼ j a ð t Þ  D t > ij :

ð22Þ

i–j;j¼1

where aij(t) is calculated by Formula (9). From the above analysis, it may be observed that the main factors affecting the transition probability are the number of evacuees, the maximum node capacity, the mass evacuation speed (reflected by the panic correction coefficient) and the toxic gas concentration. By employing the transition probability formula, changes in crowd distribution in the evacuation area may be calculated, and the randomness impact of these factors on the mass evacuation may be analyzed by adjusting the values of the relevant parameters.

3. The simulated evacuation scenario Let us suppose that an evacuation area has a radius of 720 m and is divided into 256 (16  16) grids of 90 m  90 m, so the total evacuation area is approximately 1.6 square kilometers. A terrorist CO gas attack occurs in this area and the local population in this area must be evacuated immediately. Our objective is to study the impact of certain factors on the evacuation process. The numbering rule of the grid nodes is: 8 5 n, m 6 8 and the coordinate of the center grid is (0, 0) (see Fig. 2). From the divided grid, 544 (2  16  17) connections representing the road travelling capacity between nodes are established. The initial number of population to be evacuated is 256  10 = 2560, i.e. for each node, Pi(0) = 10, representing the population distribution at time t = 0. The maximum node capacity: Mi = 20. The normal evacuation speed is 90 m/min. From the evacuation speed and the length of the grid, the quantitative relationship between time and space may be established, by calculating the time cost of crossing a certain number of nodes. Suppose that the initial concentration of the instantaneous leakage of toxic CO gas is C0, which will decay exponentially with time according to the formula: C = C0  et/10]. Then the variation trend of aij ðt Þ under the influence of toxic gas may be calculated, as illustrated in Fig. 3. It may be observed that when CO concentration is 5000 ppm, aij ðt Þ is less than 15% of that under normal circumstances; when the CO concentration decays to approximately 800 ppm, aij ðt Þ is approximately 75% of that under normal circumstances; when the CO concentration is very low, namely less than 100 ppm, then aij ðtÞ is quite close to that under normal circumstances. This variation trend of aij ðt Þ under the influence of toxic gas is closely related to the endurance capacity of the human body to toxic gas. Table 1 outlines the impact of different concentrations of CO gas on the human body. It may be noted that when CO concentration reaches 5000 ppm, people will suffer headaches and dizziness within 3–5 min, and lose consciousness within

Fig. 2. A schematic diagram of the evacuation area. Note: The upper and lower bounds of the evacuation area are designated as evacuation exits.

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20 min. Thus it is reasonable that aij ðt Þ under this concentration is less than 15% of that under normal circumstances. At a CO concentration of 800 ppm, people will suffer headaches and dizziness and vomit within 40 min, and will lose consciousness within 2 h: again, it is reasonable that aij ðtÞ under this concentration is approximately 75% of that in normal circumstances. When the CO concentration is very low and has little influence on human body, aij ðt Þ will be close to that under normal circumstances. From these data, the decaying formula of CO concentration assumed in this work appears rational, and can realistically reflect the negative influence of crowd physiological risk caused by toxic gas diffusion on evacuation route selection. 4. Results and discussion In the following discussion we will study the uncertainties in crowd route selection by analyzing the variation rule of clearance time (when the remaining population is less than 10) under the influence of relevant parameters. 4.1. Impact of initial number of population on clearance time We set the initial number of population to be evacuated at P1 = 2560, P2 = 0.5P1, P3 = 2P1, P4 = 3P1, P5 = 0.3P1 and P6 = 5P1 in different tests, and calculated the trend of remaining population over time under the conditions Mi = 20 and U = 90 m/min. Fig. 4 presents the results. It may be observed that the greater the initial number of population, the higher the number stranded after a given evacuation time. At the start of evacuation, evacuation efficiency rises within a few minutes, but as time progresses, due to the limitations of the road network capacity, the evacuation efficiency decreases significantly. Fig. 5 shows the results as logarithmic – linear coordinates: these data suggest that there is a piecewise linear relationship between the logarithm of remaining population and the evacuation time. Within a few minutes of the start of evacuation, the absolute value of the slope is larger, indicating higher evacuation efficiency. When the number of remaining population reduces to a certain value, the absolute value of the slope begins to decrease. The greater the initial number of population, the more marked the inflection point of the slope will be. When the initial number of population is lower, there is a single linear relationship between the logarithm of remaining population and the evacuation time. It should be noted that regardless of the initial number of evacuees, the slopes after the inflection point essentially stabilize at the value of the lowest number of initial evacuees. This finding verifies the repeatability of the stochastic Markov evacuation model. Table 2 and Fig. 6 further illustrate the changes in clearance time under different initial numbers of population. It may be observed that there is a kind of linear relationship between the final clearance time and the logarithm of initial number of population, i.e. y / lg x. 4.2. Impact of maximum node capacity on clearance time We set the maximum node capacity as 5, 10, 20, 30 and 50 in different tests, and calculated the trend of remaining population over time under the conditions P = 2560 and U = 90 m/min. The results are presented in Fig. 7. It may be observed that where the initial number of population to be evacuated remains the same, the larger the maximum node capacity, the greater the evacuation efficiency and the earlier the final clearance time. In the logarithmic – linear coordinates as shown in Fig. 8, a piecewise linear relationship between the logarithm of remaining population and the evacuation time is also apparent. The larger the maximum node capacity, the more evident such piecewise linear characteristics will be. For larger maximum node capacities, the slopes after the inflection point stabilize at the approximate value of the

Fig. 3. The variation trend of aij ðt Þ under the influence of toxic gas (when P i ð0Þ ¼ 10, Mi = 20, U = 90 m/min).

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Table 1 The impact of CO concentration on the human body [27]. Gas name

Gas concentration (ppm)

Impact on the human body

CO

50 200 400 800 1600 3200 6400 12,800

The OSHA permissible exposure concentration which people can endure for 8 h Individuals may suffer slight frontal headache after 2–3 h Individuals may suffer frontal headache and vomiting after 1–2 h, and vertigo after 2.2–3.5 h Individuals may suffer headache, dizziness and vomiting within 45 min, and coma, or even death within 2 h Individuals may suffer headache, dizziness and vomiting within 20 min, and coma, or even death within 1 h Individuals may suffer headache and dizziness within 5–10 min, and coma, or even death within 30 min Individuals may suffer headache and dizziness within 1–2 min, and coma, or even death within 10–15 min Individuals may fall into a coma immediately, and die within 1–3 min

Fig. 4. The trend of remaining population with time under different initial numbers of population.

Fig. 5. The trend of the logarithm of remaining population with time under different initial numbers of population.

smallest maximum node capacity, which suggests that the evacuation status of the entire road network will eventually reach a universal equilibrium, regardless of the maximum node capacity. This equilibrium may be due to the combined effect of the influence of crowd movement speed, random route selection behavior and other factors. Table 3 and Fig. 9 further illustrate the differences in clearance time under different maximum node capacities. It may be observed that final clearance time will decrease linearly with an increase in maximum node capacity, i.e. y / x.

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Table 2 Clearance time under different initial numbers of population.

Clearance time (min)

0.3 * P1

0.5 * P1

P1 = 2560

2 * P1

3 * P1

5 * P1

44

48

52

55

56

59

4.3. Impact of crowd panic on clearance time In this work, the degree of crowd panic is reflected by the correction coefficient c of the evacuation speed caused by panic emotion:

c = 1, i.e. U = 90 m/min for ideal evacuation routes. c = 1.3, i.e. U = 117 m/min for feasible evacuation routes. c = 1.8, i.e. U = 162 m/min for escape evacuation routes. We calculate the clearance time under different degrees of crowd panic when P = 2560 and Mi = 10. Fig. 10 depicts the trend of remaining population over time. It may be observed that the modification of evacuation speed according to the degree of psychological panic in the situation of evacuation on foot has little influence on the overall evacuation process, with only a minimal impact on the final

Total evacuation time

60

55 Total Evacuation Time

Total Evacuation Time

60

50

45

40 2000

4000

6000

8000

10000 12000 14000

Initial evacuation population

(a) in linear coordinates

Total evacuation time Linear fit

55

50

45

40 100

1000

10000

Initial evacuation population

(b) in linear- logarithmic coordinates

Fig. 6. The trend of clearance time under different initial numbers of population.

Fig. 7. The trend of remaining population over time under different maximum node capacities.

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Fig. 8. The trend of the logarithm of remaining population over time under different maximum node capacities.

Table 3 Clearance time under different maximum node capacities.

Clearance time (min)

0.25Mi

0.5Mi

Mi = 20

1.5Mi

2.5Mi

57

54

52

47

43

Total Evacuation Time

Total evacuation time Linear fit

50

40

30 0

10

20

30

40

50

Mi Fig. 9. The trend of clearance time under different maximum node capacities.

clearance time. That is to say, faster evacuation speeds may somewhat reduce the final clearance time. In the logarithmic – linear coordinates, as shown in Fig. 11, there is also a piecewise linear relationship between the logarithm of remaining population and evacuation time, which suggests the evacuation efficiency is higher at the outset, but decreases with time. There is little difference between these piecewise linear characteristics at different evacuation speeds. This may be because the evacuation process is affected by a variety of factors, including node capacity and population distribution, thus the modification here within a small range of values (across the range of pedestrian evacuation speed) due to evacuation speed may be not apparent in the stochastic evacuation model as a whole. 4.4. Impact of the crowd physiological risk on clearance time From the analysis in Section 4, we assume that the initial concentration of the instantaneously released toxic gas CO is C0 = 10,000 ppm, which will decay exponentially with time, as C = C0  et/10. We set the initial parameters as P = 2560, Please cite this article in press as: J. Wang et al., Randomness in the evacuation route selection of large-scale crowds under emergencies, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.033

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Fig. 10. The trend of remaining population over time under different degrees of crowd panic.

Fig. 11. The trend of the logarithm of remaining population over time under different degrees of crowd panic.

Fig. 12. The trend of remaining population over time with and without consideration of the physiological risk caused by toxic gas.

Mi = 20 and U = 90 m/min. Fig. 12 presents a comparison of the remaining population when the physiological risk caused by toxic gas is taken into consideration with the remaining population level under normal circumstances. It may be observed that in the same evacuation scenario, when the physiological risk caused by toxic gas is not considered, the clearance time is 52 min, whereas when the physiological risk is considered, the clearance time increases to 83 min. This difference reflects the importance of the consideration of crowd physiological risk in large-scale evacuation, when faced with a disaster such as toxic gas diffusion.

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Fig. 13. The proportion of population remaining at t = 30 min with and without consideration of crowd physiological risk.

Given the decay function of CO gas, the concentration of CO will be 500 ppm at t = 30 min. From the time when the toxic gas begins to diffuse until the concentration decreases to 500 ppm, the cumulative exposure to toxic gas is sufficient to cause the evacuation to fail, i.e. people may die or be seriously injured. Let us compare the proportion of population remaining at t = 30 min in this scenario with same population under normal circumstances, as illustrated in Fig. 13. Members of the evacuating crowd remaining at t = 30 min are thought to be at high risk of fatality in such a toxic gas atmosphere: in this stochastic evacuation model, only 3.23% of the entire crowd is at high risk of fatality without consideration of the physiological risk caused by toxic gas, while after modification according to crowd physiological risk, 24.44% of the total crowd population is at high risk of fatality at t = 30 min. Evidently, the latter result is more realistic in an emergency evacuation plan and will lead to a higher safety threshold. 5. Conclusions In this work, we focuses on the evacuation crowd, and introduces and quantifies two subjective factors i.e. crowd physiological and psychological factor, to establish and modify a random Markov route selection model of the evacuees integrated with other key parameters, such as the initial number of population to be evacuated and the maximum node capacity. The Markov process probabilistic description and resulting uncertainties of the evacuation process are then analyzed in detail under a background of instantaneous leakage and diffusion of CO poison gas. Through application of the variation rule of clearance time under the influence of relevant parameters, it was determined that: (1) The efficiency of large-scale stochastic evacuation is closely related to the initial number of population (the crowd scale) and the maximum node capacity. With increases in crowd scale, an increasingly marked piecewise linear relationship appears between the logarithm of remaining population and evacuation time, and the absolute value of the slope reduces from high to low, reflecting a change from high to low in evacuation efficiency. Moreover, final clearance time increases linearly with an increase in the logarithm of the initial number of population, i.e. y / lg x. (2) With increases in maximum node capacity, the logarithm of remaining population and evacuation time also present an increasingly marked piecewise linear relationship, while final clearance time decreases linearly with the increase in maximum node capacity, i.e. y / x. (3) Modification of evacuation speed within a small range of values (the range of walking speeds) according to the degree of psychological panic in an evacuation on foot has little influence on the overall evacuation process. (4) In the same evacuation scenario, when the physiological risk caused by toxic gas is not considered, the clearance time and the proportion of remaining population at high risk to life at t = 30 min are much less than that when physiological risk is considered. Thus, physiological risk under an atmosphere of poison gas, for example, can markedly affect the evacuation results and should be carefully considered in large-scale emergency evacuation.

Acknowledgement This work was supported by the National Natural Science Foundations of China (Nos. 91024025, 21406115 and 2014M551580), the Natural Science Foundation of Jiangsu Province (Nos. BK20140950 and 14KJB620001), the Postdoctoral Research Funding Plan of Jiangsu Province (No. 1302052B) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The authors deeply appreciate the supports. Please cite this article in press as: J. Wang et al., Randomness in the evacuation route selection of large-scale crowds under emergencies, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.033

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